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DOI: 10.23671/VNC.2018.4.23390
Convergence of the Lagrange-Sturm-Liouville Processes for Continuous Functions of Bounded Variation
Trynin, A. Y.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 4.
Abstract: The uniform convergence within an interval \((a,b)\subset [0,\pi]\) of Lagrange processes in eigenfunctions \(L_n^{SL}(f,x)=\sum\nolimits_{k=1}^{n}f(x_{k,n})\frac{U_n(x)}{U_{n}'(x_{k,n})(x-x_{k,n})}\) of the Sturm-Liouville problem is established. (Here \(0<x_{1,n}<x_{2,n}<\dots<x_{n,n}<\pi\) denote the zeros of the eigenfunction \(U_n\) of the Sturm-Liouville problem.) A continuous functions \(f\) on \([0,\pi]\) which is of bounded variation on \((a,b)\subset [0,\pi]\) can be uniformly approximated within the interval \((a,b)\subset [0,\pi]\). A criterion for uniform convergence within an interval \((a,b)\) of the constructed interpolation processes is obtained in terms of the maximum of the sum of the moduli of divided differences of the function \(f\). Outside the interval \((a, b)\), the Lagrange interpolation process may diverge. The boundedness in the totality of the Lagrange fundamental functions constructed from eigenfunctions of the Sturm-Liouville problem is established. The case of the regular Sturm-Liouville problem with a continuous potential of bounded variation is also considered. The boundary conditions for the third kind Sturm-Liouville problem without Dirichlet conditions are studied. In the presence of service functions for calculating the eigenfunctions of the regular Sturm-Liouville problem, the Lagrange-Sturm-Liouville operator under study is easily implemented by computer technology.
For citation: Trynin, A. Y. Convergence of the Lagrange-Sturm-Liouville Processes for Continuous Functions of Bounded Variation, Vladikavkaz Math. J., 2018, vol. 20, no. 4, pp. 76-91 (in Russian). DOI 10.23671/VNC.2018.4.23390
1. Kramer, H. P. A Generalized Sampling Theorem, J. Math. Phus., 1959, vol. 38, pp. 68-72.
2. Natanson, G. I. Ob odnom interpolyatsionnom protsesse, Uchen. zapiski Leningrad. ped. in-ta im. A. I. Gertsena, 1958, vol. 166, pp. 213-219. (in Russian).
3. Novikov, I. Ya. and Stechkin, S. B. Basic Wavelet Theory, Russian Mathematical Surveys, 1998, vol. 53, no. 6, pp. 1159-1231. DOI: 10.1070/RM1998v053n06ABEH000089.
4. Novikov, I. Ya. and Stechkin, S. B. Basic Constructions of Wavelets, Fundamental and Applied Mathematics, 1997, vol. 3, no. 4, pp. 999-1028. (in Russian).
5. Stenger, F. Numerical Metods Based on Sinc and Analytic Functions, N.Y.,
Springer, 1993, 565 p.
6. Dobeshi, I. Ten Lectures on Wavelets, Society for Industrial and Appled Mathematics, Philadelphia, Pennsylvana, 1992.
7. Oren E. Livne and Achi E. Brandt. MuST: the Multilevel Sinc Transform, SIAM Journal on Scientific Computing, 2011 vol. 33, no. 4, pp. 1726-1738. DOI: 10.1137/100806904.
8. Coroianu, L. and Sorin, G. Gal. Localization results for the non-truncated
max-product sampling operators based on Fejer and sinc-type kernels,
Demonstratio Mathematica, 2016, vol. 49, no. 1, pp. 38-49.
9. Richardson, M. and Trefethen, L. A Sinc Function Analogue of Chebfun,
SIAM Journal on Scientific Computing, 2011, vol. 33, no. 5, pp.2519-2535. DOI: 10.1137/110825947.
10. Khosrow, M., Yaser, R. and Hamed, S. Numerical Solution for First Kind Fredholm Integral Equations by Using Sinc Collocation Method, International Journal of Applied Physics and Mathematics, 2016, vol. 6, no. 3, pp. 88-94. DOI: 10.17706/ijapm.2016.6.3.88-94.
11. Trynin, A. Yu. Necessary and Sufficient Conditions for the Uniform on a Segment Sinc-Approximations Functions of Bounded Variation, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., vol. 16, no. 3, pp. 288-298 (in
Russian). DOI: 10.18500/1816-9791-2016-16-3-288-298.
12. Trynin, A. Yu. and Sklyarov, V. P. Error of Sinc Approximation of Analytic Functions on an Interval, Sampling Theory in Signal and Image Processing, 2008, vol. 7, no 3, pp. 263-270.
13. Marwa M. Tharwat. Sinc Approximation of Eigenvalues of Sturm-Liouville Problems with a Gaussian Multiplier, Calcolo: a Quarterly on Numerical Analysis and Theory of Computation, 2014, vol. 51, no. 3, pp. 465-484. DOI: 10.1007/s10092-013-0095-3.
14. Trynin, A. Yu. On the Estimation of the Approximation of Analytic Functions by an Interpolation Operator with Respect to Syncs, Matematika. Mekhanika [Mathematics. Mechanics], Saratov, Saratov Univ., 2005, vol. 7, pp. 124-127. (in Russian).
15. Zayed, A. I. and Schmeisser, G. New Perspectives on Approximation and Sampling Theory, Applied and Numerical Harmonic Analysis, N.Y.-Dordrecht-London, Springer Int. Publ. Switzerland, 2014. DOI: 10.1007/978-3-319-08801-3.
16. Trynin, A. Yu. Estimates for the Lebesgue Functions and the Nevai Formula for the Sinc-Approximations of Continuous Functions on an Interval, Siberian Mathematical Journal, 2007, vol. 48, no. 5, pp. 929-938. DOI: 10.1007/s11202-007-0096-z.
17. Trynin, A. Yu. Tests for Pointwise and Uniform Convergence of Sinc Approximations of continuous functions on a Closed Interval, Sbornik: Mathematics, 2007, vol. 198, no. 10, pp. 1517-1534. DOI: 10.1070/SM2007v198n10ABEH003894.
18. Trynin, A. Yu. A Criterion for the Uniform Convergence of Sinc-Approximations on a Segment, Russian Mathematics (Iz. VUZ), 2008, vol. 52, no. 6, pp. 58-69. DOI: 10.3103/S1066369X08060078.
19. Sklyarov, V. P. On the best uniform sinc-approximation on a finite interval,
East Journal on Approximations, 2008, vol. 14, no. 2, pp. 183-192.
20. Trynin, A. Yu. On Divergence of Sinc-Approximations Everywhere on \((0,\pi)\), St. Petersburg Mathematical Journal, 2011, vol. 22, pp. 683-701. DOI: 10.1090/S1061-0022-2011-01163-X.
21. Trynin, A. Yu. On Some Properties of Sinc Approximations of Continuous Functions on the Interval, Ufa Mathematical Journal, 2015, vol. 7, no. 4, pp. 111-126. DOI: 10.13108/2015-7-4-111.
22. Umakhanov, A. Y. and Sharapudinov, I. I. Interpolation of Functions by the Whittaker Sums and Their Modifications: Conditions for Uniform Convergence, Vladikavkaz Math. J., 2016, vol. 18, pp. 61-70. DOI: 10.23671/VNC.2016.4.5995.
23. Umakhanov, A. Y. and Sharapudinov, I. I. Interpolation of Functions by Whittaker Sums and their Modifications: Conditions of Uniform Convergence, Materialy 18th Intern. Saratov Winter School. "Modern Problems of the Theory of Functions and their Applications", Saratov, 2016, pp. 332-334.
24. Trynin, A. Yu. On Necessary and Sufficient Conditions for Convergence of Sinc-Approximations, St. Petersburg Mathematical Journal, 2016, vol. 27, no. 5, pp. 825-840. DOI: 10.1090/spmj/1419.
25. Trynin, A. Yu. Approximation of Continuous on a Segment Functions with the Help of Linear Combinations of Sincs, Russian Mathematics (Iz. VUZ), 2016, vol. 60, no. 3, pp. 63-71. DOI: 10.3103/S1066369X16030087.
26. Trynin, A. Yu. A Generalization of the Whittaker-Kotel'nikov-Shannon Sampling Theorem for Continuous Functions on a Closed Interval, Sbornik: Mathematics, 2009, vol. 200, no. 11, pp. 1633-1679. DOI: 10.1070/SM2009v200n11ABEH004054.
27. Trynin, A. Yu. On Operators of Interpolation with Respect to Solutions of a Cauchy Problem and Lagrange-Jacobi Polynomials, Izvestiya: Mathematics, 2011, vol. 75, no. 6, pp. 1215-1248. DOI: 10.1070/IM2011v075n06ABEH002570.
28. Trynin, A. Yu. On the Absence of Stability of interpolation in Eigenfunctions of the Sturm-Liouville Problem, Russian Mathematics (Iz. VUZ), 2000, vol. 44, no. 9, pp. 58-71.
29. Trynin, A. Yu. Differential Properties of Zeros of Eigenfunctions of the Sturm-Liouville Problem, Ufa Mathematical Journal, 2011, vol. 3, no. 4, pp. 130-140.
30. Trynin, A. Yu. On Inverse Nodal Problem for Sturm-Liouville Operator, Ufa Mathematical Journal, 2013, vol. 5, no. 4, pp. 112-124. DOI: 10.13108/2013-5-4-112.
31. Trynin, A. Yu. The Divergence of Lagrange Interpolation Processes in Eigenfunctions of the Sturm-Liouville Problem, Russian Mathematics (Iz. VUZ), 2010, vol. 54, no. 11, pp. 66-76. DOI: 10.3103/S1066369X10110071.
32. Trynin, A. Yu. Localization Principle for Lagrange-Sturm-Liouville Processes, Matematika. Mekhanika [Mathematics. Mechanics], Saratov, Saratov Univ., 2005, vol. 8, pp. 137-140 (in Russian).
33. Trynin, A. Yu. On one Integral Sign of Convergence of Lagrange-Sturm-Liouville Processes, Matematika. Mekhanika [Mathematics. Mechanics], Saratov, Saratov Univ., 2007, vol. 9, pp. 94-97.
34. Trynin, A. Yu. Teorema otschetov na otrezke i ee obobscheniya, LAP LAMBERT Acad. Publ., 2016, 488 p. (in Russian).
35. Golubov, B. I. Spherical Jump of a Function and the Bochner-Riesz Means of Conjugate Multiple Fourier Series and Fourier Integrals, Mathematical Notes, 2012, vol. 91, no. 3-4, pp. 479-486. DOI: 10.4213/mzm8739.
36. Dyachenko, M. I. On a Class of Summability Methods for Multiple Fourier Series, Sbornik: Mathematics, 2013, vol. 204, no. 3, pp. 307-322. DOI: 10.4213/sm8118.
37. Maksimenko, I. E. and Skopina, M. A. Multidimensional Periodic Wavelets, St. Petersburg Mathematical Journal, 2004, vol. 15, no. 2, pp. 165-190. DOI: 10.1090/S1061-0022-04-00808-8.
38. Dyachenko, M. I. Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series, Mathematical Notes, 2004, vol. 76, no. 5, pp. 673-681. DOI: 10.4213/mzm139.
39. Borisov, D. I. and Dmitriev, S. V. On the Spectral Stability of Kinks in 2D Klein-Gordon Model with Parity-Time-Symmetric Perturbation, Studies in Applied Mathematics, 2017, vol. 138, no. 3, pp. 317-342.
40. Golubov, B. I. Absolute Convergence of Multiple Fourier Series, Mathematical Notes, 1985, vol. 37, no. 1, pp. 8-15. DOI: 10.1007/BF01652507.
41. Borisov, D. I. and Znojil, M. On Eigenvalues of a \(\mathscr{PT}\)-Symmetric Operator in a thin Layer, Sbornik: Mathematics, 2017, vol. 208, no. 2, pp. 173-199. DOI: 10.1070/SM8657.
42. Ivannikova, T. A., Timashova, E. V. and Shabrov, S. A. On Necessary Conditions for a Minimum of a Quadratic Functional with a Stieltjes Integral and Zero Coefficient of the Highest Derivative on the Part of the Interval, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2013, vol. 13, no. 2(1), pp. 3-8. (in Russian).
43. Farkov, Yu. A. On the Best Linear Approximation of Holomorphic Functions, Journal of Mathematical Sciences, 2016, vol. 218, no. 5, pp. 678-698. DOI: 10.1007/s10958-016-3050-4.
44. Sansone, D. Obyknovennye differencial'nye uravneniya [Ordinary Differential Equations], Moscow, Foreign Languages Publ. House, 1953, vol. 1, 2.